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Séminaire Lotharingien de Combinatoire, 86B.35 (2022), 12 pp.

# Amritanshu Prasad and Samrith Ram

# Set Partitions, Tableaux, and Subspace Profiles of Regular Diagonal Operators

**Abstract.**
We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At -1, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express *q*-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. In a special case these polynomials coincide with those defined by Touchard in his study of crossings of chord diagrams.

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

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